Methods of forecast verification

Transcription

1 Methods of forecast verification Kiyotoshi Takahashi Climate Prediction Division Japan Meteorological Agency 1

2 Outline 1. Purposes of verification 2. Verification methods For deterministic forecasts For probabilistic forecasts Standardised Verification System for Long-Range Forecasts (SVSLRF) 3. Results on the TCC web page 2

3 1. Purposes of verification Forecast verification is a process of assessing quality of forecasts. to monitor forecast quality how accurate are forecasts and are they improving? to guide forecasters and users help forecasters understand model characteristics help us provide higher value forecasts to users to guide future developments of system identify model faults and improve systems help us compare and evaluate different forecasts (model or guidance) 3

4 2. Verification methods For deterministic forecast ACOR, RMSE, Bias (ME),MSSS (ROC) For probabilistic forecast ROC, Reliability diagram, BSS (Brel, Bres) 4

5 Root Mean Square Error (RMSE) 1 RMSE N N F i O i i1 Range: 0 to infinity, Perfect score: 0. 2 F : forecast O : observation N :sample size RMSE measures absolute magnitude of the forecast error. It does not indicate the direction the error. RMSE small RMSE large 5

6 Mean Error (ME) ME 1 N N F i O i i1 F : forecast O : observation N :sample size Range: variable, Perfect score: 0. ME measures average magnitude of the forecast error. It indicates the direction the error. RMSE N N i ME e e xi ai ME RMSE can be divided into ME(systematic error) and random error ( ). e 2 6

7 Anomaly Correlation (AC) Range: -1 to 1. Perfect score: 1. AC measures correspondence or phase difference between forecast and observation, subtracting out the climatological mean at each point. N i i i N i i i N i i i i i C O C F C O C F AC ) ( ) ( ) )( ( climatology observation forecast : : : C O F 7 AC <<1 AC 1

8 AC positive ( 1) RMSE large AC negative RMSE large AC positive( 1) RMSE small AC negative RMSE small 8

9 Mean Squared Skill Score (MSSS) MSSS 1 MSE MSE c Perfect score: 1 (when MSE=0) Climatology forecast score: 0 where MSE is the mean squared error MSE 1 N N F i O i i1 2 F : forecast O : observation and MSE c is the MSE of climatology forecast. MSSS can be expanded (Murphy,1988) as 2 2 s f s f f o MSSS 2 r fo 2 so so so 2n 1 2n n 1 n 1 s : squared root of variance r : correlation f : forecast o : observation n 1 Fi F O i O n i1 rfo The first 3 terms are related to S f SO 1 phase error (through the correlation) 2amplitude errors (through the ratio of the forecast to observed variances) 3bias error 9

10 Reliability diagram The reliability diagram plots the observed frequency(y-axis) against the forecast probability(x-axis). The diagonal line indicates perfect reliability (observed frequency equal to forecast probability for each category). climatology perfect reliability Brier Scores forecast frequency Points below(above) the diagonal line indicate overforecasting (underforecasting). 10

11 BS 1 N Brier (skill) score Brier score measures mean squared error of the probability forecasts. N i1 ( p i o i 2 ) Range: 0 to 1. Perfect score: 0 Random:1/3 p o i i N : forecast : observed : sample size probability Brier skill score measures skill relative to a reference forecast (usually climatology). BSS 1 BS BS reference Climatology: occurrence(0or 1) o( 1 o) Range: minus infinity to 1. BSS=0 indicates no skill when compared to the reference forecast. Perfect score: 1. 11

12 Decomposition of the Brier score Murphy(1973) showed that the Brier score could be partitioned into three terms(for K probability classes and N samples). These terms are shown separately to attribute sources of error. o :climatological occurrence BS 1 K K 1 2 n ( p o ) n ( o o) 2 k k k k k N N k1 k1 o(1 o) reliability (brel) resolution (bres) uncertainty (bunc) the mean squared difference between the forecast probability and the observed frequency. Perfect score: 0 the mean squared difference between the observed frequency and climatological frequency. - indicates the degree to which the forecast can separete different situations. climatologial forecast score:0 Perfect score: o( 1 o) measures the variability of the observations. 12

13 Brier skill score Brier skill score = the relative skill of the probabilistic forecast to the climatology BSS BS BS BS BS clim 1 0 BSc lim clim Reliability skill score brel Brel 1 BS c lim Skill score BS c bunc lim score score Range: minus infinity to 1. Perfect score: 1 BSS=0 indicates no skill when compared to the climatology. BSS>0 : better than clim. Resolution skill score Bres bres BS clim Perfect score: 1 Perfect score: 1 forecast perfect forecast score score reference reference Score 100 The larger these skill scores are, the better. 13

14 Interpretation of Reliability diagram and BSS Event : Z500 Anomaly > 0 Northern Hemisphere Spring of 2008 (2008/2/28 ~2008/5/29) First week forecast (day 2-8) Third and fourth week (day 16-29) overforecasting underforecasting BSS>0 better than climatology BSS<0 inferior to climatology 14

15 Relative Operating Characteristic (ROC) ROC is created by plotting the hit rate(y-axis) against the false alarm rate(x-axis) using increasing probability thresholds to make the yes/no decision. The area under the ROC curve (=ROC area) is frequently used as a score. 15

16 Forecast Steps for making ROC diagram 1. For each forecast probability category, count the number of hits,misses, false alarms, and correct non-events 2. Compute the hit rate and false alarm rate in each category k hit rate k = hits k / (hits k + misses k ) false alarm rate k = false alarms k / (false alarms k + correct non-events k ) 3. Plot hit rate vs false alarm rate 4. ROC area is the integrated area under the ROC curve yes no yes hits false alarms no misses correct nonevents total Observed Observed yes Observed no Contingency Table 16

17 Interpretation of ROC curves Event: Z500 anomaly > 0 Spring st week forecast 3rd and 4th week forecast Perfect performance high resolution (high potential skill) low resolution (low potential skill) ROC is not sensitive to bias in forecasts. Forecasts with bias may have a good ROC curve if resolution is still good. In this sense, the ROC can be considered as a measure of potential usefulness. On the other hand, reliability diagram is sensitive to the bias. It is needed to see both the ROC and the reliability diagram. 17

18 SVSLRF (Standard Verification System for Long-Range Forecast) WMO standard tool to verify skill in seasonal models It was introduced by the Commission for Basic Systems (CBS) of the World Meteorological Organization (WMO) in December, Users can appropriately evaluate forecast skill with common measures. 18

19 Outline of SVSLRF Mandatory part Parameters Verification regions Deterministic forecasts Probabilistic forecasts Level 1 T2m anomaly Tropics(20S-20N) MSSS ROC curves Precipitation anomaly (Nino3.4 Index) Northern extratropics(20n-90n) Southern extratropics(20s-90s) (N/A) ROC areas Reliability diagrams Frequency histograms Level 2 T2m anomaly Precipitation anomaly (SST anomaly) Grid-point verification on a 2.5 by 2.5 grid MSSS and its three-term decomposition at each gridpoint ROC areas at each grid-point Level 3 T2m anomaly Precipitation anomaly (SST anomaly) Grid-point verification on a 2.5 by 2.5 grid 3 by 3 contingency tables at each grid-point ROC reliability tables at each grid-point 19

20 LC-SVSLRF Lead Centre for the Long-Range Forecast Verification System Australian Bureau of Meteorology (BOM) Meteorological Service of Canada (MSC) URL: The Manual is here Lead Center web site You can get scores of several centers on Lead Centre website. 20

21 3. Verification results on TCC web page 1-month forecast 3-month forecast Warm/Cold season forecast 21

22 Verification of operational 1-month forecasts Deterministic forecast Ensemble mean forecast error maps, RMSE and Anomaly Correlation Probabilistic forecast Time sequence of RMSE and AC 22

23 Verification of 1-month Ensemble mean forecast maps (Deterministic) Z500 over the Northern Hemisphere Observation Forecast Error Stream function (850, 200hPa) RMSE and Anomaly Corr

24 Verification of 1-month Ensemble mean forecast maps (Deterministic) Z500 over the Northern Hemisphere Observation Forecast Error ANL-Clim (JRA-25/JCDAS) FCT-Clim (model) 24

25 Verification of operational 1-month forecasts Probabilistic forecast Reliability diagrams and Brier skill scores ROC curves and area for each season 25

26 Verification of hindcast based on SVSLRF 26

27 Hindcast verification methods based on Standardised Verification System for Long-Range Forecasts (SVSLRF) Verification of deterministic forecasts Mean Square Skill Score (MSSS) Contingency tables Verification of probabilistic forecasts Reliability diagrams ROC curves and ROC areas 27

28 Examples of MSSS for precipitation Dec-Feb started on 10 Nov. Jun-Aug started on 10 May. Positive MSSS indicates that the forecast is better than climatological forecast. 28

29 Deterministic Summary Index Random Climatology Perfect RMSE >0 0 ME 0 AC MSSS 0 +1 MSE >0 >0 0 Probabilistic Index Random Climatology Perfect Reliability diagram o( 1 o) Fit to the diagonal line BS 1/3 bunc= 0 BSS(x100) 100> Brel(x100) 100> 100> +100 Bres(x100) 100> Roc area(x100) 100>

30 References Murphy, A.H., 1973: A new vector partition of the probability score. J. Appl. Meteor., 12, Murphy, A.H., 1988: Skill scores based on the mean square error and their relationships to the correlation coefficient. Mon. Wea. Rev., 16,